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Use a reversible circuit $T$ to compute an $L=O(n)$-bit representation of the angle into a workspace register, with all additional work bits retained:Write that representation as ${\theta }_x={\sum }_{l=1}^L{b}_l(x){\alpha }_l$, with known binary place values ${\alpha }_l$ (including the fixed scale and any integer bit). On the target, apply a controlled unitary gate $R_y(2{\alpha }_l)$ controlled by each angle bit $b_l$. Rotations about the same axis add their angles, so their product isFinally apply $T^{†}$ to perform uncomputation. The workspace returns to zero while the control register and rotated target remain unchanged; the construction works coherently on every superposition of $x$.

A polynomial time classical calculation has a polynomial-size reversible circuit with Toffoli gates, and each Toffoli gate has a constant-size decomposition into one-qubit and two-qubit gates. The $L$ controlled rotations are themselves two-qubit gates. Thus the total quantum circuit size is $\mathrm{poly}(n)$. This is the binary-angle implementation of a quantum variable rotation; the stipulated angle precision is understood.

The one-qubit rotation gatesatisfies $R_y(2\theta )|0⟩=\cos \theta |0⟩+\sin \theta |1⟩$ and $R_y(\alpha )R_y(\beta )=R_y(\alpha +\beta )$. The latter identity enables a binary-angle implementation of a quantum variable rotation.